Conjecture 3.3: Definition and characterization of σ via Ψ for Weyl groups

Establish that for any finite Weyl group W, the following hold for the map defined using the polynomials Ψ_{C,E} ∈ Z[v] from cl(W) to Z[v][Irr(W)]: (1) For each conjugacy class C ∈ cl(W), letting Irr*(W) = {E ∈ Irr(W) : there exists C with Ψ_{C,E} ≠ 0 and the leading coefficient of Ψ_{C,E} negative}, X_C = {E ∈ Irr(W) \ Irr*(W) : Ψ_{C,E} ≠ 0}, and b_E the minimal n such that E occurs in the n-th symmetric power of the reflection representation ρ of W, the subset X_C^max = {E ∈ X_C : b_{E'} ≤ b_E for all E' ∈ X_C} consists of a single element σ(C); this defines a map σ : cl(W) → Irr(W). (2) Show that σ(C) equals the geometric map Φ(C) from conjugacy classes to irreducible representations, and that the image of σ equals Irr(W) \ Irr*(W). (3) Prove that for every C ∈ cl(W), Ψ_{C,σ(C)} = v^{|w| + m(w) − m(1)}, where w ∈ C_min is any element of minimal length in C, m(w) is the multiplicity of the eigenvalue 1 of w on ρ, and m(1) = dim ρ.

Background

Section 3 proposes a new, partly conjectural, representation-theoretic definition of a map σ : cl(W) → Irr(W) for a finite Weyl group W that is designed to coincide with the previously defined geometric map Φ from conjugacy classes in W to irreducible representations, which parametrizes strata in a connected reductive group with Weyl group W. This construction is based on the polynomial-valued map Ψ: cl(W) → Z[v] [Irr(W)] factored through matrices built from Hecke algebra data and Green functions.

Given Ψ, the authors define the exceptional subset Irr*(W) ⊂ Irr(W) consisting of representations E for which some Ψ{C,E} has negative leading coefficient. For each conjugacy class C, the set X_C ⊂ Irr(W) \ Irr*(W) collects those E with Ψ{C,E} ≠ 0, and X_C^max consists of those with maximal b_E, where b_E is the minimal symmetric power degree in which E appears in the reflection representation. Conjecture 3.3 asserts that X_C^max is always a singleton, thereby defining σ(C), that σ agrees with Φ and has image Irr(W) \ Irr*(W), and that Ψ_{C,σ(C)} has the precise monomial form v^{|w|+m(w)−m(1)} for any minimal-length representative w ∈ C_min. Theorem 3.5 verifies the conjecture for several types using computation, and it also follows for rank ≤ 2 from [LY], but the general case remains conjectural as stated.